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Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q − 1, or n = 2 and q odd, we construct a connected q-regular edge-but not vertex-transitive graph of order 2q n+1. This graph is defined via a system of equations over the finite field of q elements. For n = 2 and q = 3, our graph is isomorphic to the Gray graph.

Let q be a prime power and k ≥ 2 be an integer. In [2] and [3] it was determined that the number of components of certain graphs D(k, q) introduced in [1] is at least q t−1 where t = k+2 4. This implied that these components (most often) provide the best-known asymptotic lower bound for the greatest number of edges in graphs of their order and girth. In… (More)

Let q be a prime power, F q be the field of q elements, and k; m be positive integers. A bipartite graph G ¼ G q ðk; mÞ is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over F q , with two vertices ðp 1 ; p 2 Þ 2 P and ½ l 1 ; l 2 2 L being adjacent if and only if p 2 þ l 2 ¼ p k 1 l m

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